Is 8 And A <sub>, 1 </ Sub> = 1 Distance-regular Graphs

In this thesis, by means of intersection diagrams and circuit chasing techniques, we give a classification of distancc-regular graphs with k = 8 and α1-1, and get the results as follows.Let I' be a distance-regular graph with k=8, α1=1 and r=r(Γ).1. If c_T+1=3, then d≤ 2r + 1 and one of the following holds. (1) a_r+1 = 5 and d = r + 1.(2) a_r+1=4, c_d = 8 and r + 2 ≤ d = r + t + 2 ≤ 2r + 1. (3) ra_r+1= 3, c_d = 4, Γ is a 1-homogeneous graph and r 2≤ d- r + s + 1 ≤ 2r + 1.2. If c_r+1 = 2, then one of the following holds. (1) a_r+1=6 and d=r + 1.(2)(c_r+1, a_r+1,b_r+1=(2,5,1), d = r + 2 or d = r + t + 2 ≤ 2r+l, c_r+2= 6 and c_d = 8(3)(c_r+1,a_r+1,b_r+1 = (2,4,2).(4)(c_r+1,a_r+1,b_r+1) (2,3, 3), d = r + 2 or when d > r + 3, c_r+2 = 3 or 4.(5) (c_r+1,a_r+1,b_r+1) = (2,2,4).Particularly, when (c_r+1, a_r+1, b_r+) = (2,2,4), one of the following holds,(1) c_r+1, = 4, d = r + 2 and Γ is a 1-homogeneous graph.(ii) (c_r+2,a_r+2,b_r+2) = (3,4, 1), c_r+3=... = c_d-1 =6, cd = 8 and r + 3 ≤ d=r+t₁ + 3 ≤ 2r + 2.(iii) (c_r+2, a_r+2,6_r+2 = (3,3,2), c_d = 4, Γ is a 1-homogeneous graph and r+ 3 < d - r + t + 2 < 2r + 2,3. If c_r+1= 1, then the following hold.(1)If(c_r+1,a_r+1,b_r+1)≠(1,5, 2),then c_r+2 / 1, Moreover, s=1.(2) If (c_r+1,a_r+1,b_r+1) = (1,5,2) and every c_r+2-graph is a cociiquc, then d=r + 2 or s ≥ 2.